How insert vertex in face?
Use poke faces

From the blender manual:


  Splits each selected faces into a triangle fan, create a new center vertex and create triangles between original face edges and new center vertex. The Offset can be used to make spikes or depressions.




You can then choose to offset the newly created vertex.



In versions previous to blender 2.8 the shortcut is &#9095; Alt + P.